tag:blogger.com,1999:blog-357193972024-03-07T03:49:23.724-05:00A Miscellanylogic, a spice rack, and some other thingsmackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.comBlogger63125tag:blogger.com,1999:blog-35719397.post-48706690035853423012024-01-26T19:50:00.002-05:002024-01-26T19:50:24.255-05:00Bailout Monopoly<p>Here are some changes to the rules of Monopoly that I think will make the game more fun. The first two are not new. Everyone I know already plays with them:</p><p></p><ol style="text-align: left;"><li>All required payments to the bank for taxes, Chance and Community Chest go to the center of the board.</li><li>When a player lands on Free Parking, they get all of the money on the center of the board.</li></ol><p></p><p>The next five changes are new:</p><p></p><ol start="3" style="text-align: left;"><li> Any amount of money a player is required to pay for rent, tax, Chance, or a Community Chest, that they can't pay with their own cash or with money gotten by mortgaging property, is paid by the bank.</li><li>A player always receives the rents for their property, including additional rent for houses and hotels, and doubled rents for monopolies, even if their property is mortgaged.</li><li>A player does not have to sell houses or hotels back to the bank before they can mortgage their property.</li><li>All money paid, when purchasing properties, houses and hotels from the bank, goes to the center of the board, instead of the bank.</li><li>The game ends when the bank runs out of money. Whoever has the most assets - cash plus property plus houses plus hotels minus mortgage debt - wins.</li></ol><p></p><p>The last change is also new, but I am not sure if it is necessary yet:</p><p></p><ol start="8" style="text-align: left;"><li>Some of the money in the bank - maybe a few thousand dollars - is removed at the beginning of the game so the bank runs out of money sooner. This would prevent the game from going on for too long, or from one player accumulating enough money that it is clear that the other players can't win.</li></ol><p></p><p> Apart from these changes, the game should be played with the standard rules, as written. If a player owes money and has unmortgaged property, they must pay as much of the debt as they can by mortgaging property, but Rule 3 implies that a player doesn't have to sell houses and hotel back to the bank. Players may buy, sell or trade properties amongst each other. A player may not build houses on a property unless all properties of the same color are not mortgaged. Once all houses and hotels have been purchased from the bank, no more may be bought.</p><p>Because of Rule 3, I'm calling Monopoly with these rules "Bailout Monopoly". Probably the worst part of traditional Monopoly is the slow, inexorable loss of assets that you experience when you are losing. Rule 3 prevents that from happening. If you can't pay, the bank bails you out. Also, no one can go bankrupt, so everyone gets to stay in the game until the end.</p><p>The most fun part of Monopoly, in my opinion, is collecting rent - especially when you have a monopoly and houses. With Rule 4, you can always collect rent on your property, and because of Rule 6, you don't have to give up houses once you've built them.</p><p>Rules 1, 2, 6 mean that a lot of money can be gotten by chance. That, and the luck normally involved in landing on properties, will make it impossible, at least in some games, to predict who will end up with the most money, and therefore who will win, until the end of the game.</p><p>There you have it. If you try it and like it, maybe leave a comment. Or leave a comment if you don't like it, or if you think of an improvement.</p><div><br /></div>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-29085763562884583422023-02-10T19:41:00.000-05:002023-02-10T19:41:38.397-05:00A Way to Apply Bayes' Rule to Conflicting EvidenceHere's an application of Bayes' Theorem that, IMO, deserves to be more widely known:<div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7zkOrpLp8xHDGfFp0Ojel5bn0ry-4Zq5G_24p5AbosWc-5lzqrPw72lJMh2YrJPDs7Pas1VbnAE8bz9_vR22Y7mbb6vWuNCIVXBYTYlkVTsRJ_XOA7MMZ9AppL0JbumVAiB1pYju8pKu-P8IfgcOm19SLV4dJ4m2FS1ckD0A31U6tW5_2ZA/s351/CombiningEvidence.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="120" data-original-width="351" height="109" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7zkOrpLp8xHDGfFp0Ojel5bn0ry-4Zq5G_24p5AbosWc-5lzqrPw72lJMh2YrJPDs7Pas1VbnAE8bz9_vR22Y7mbb6vWuNCIVXBYTYlkVTsRJ_XOA7MMZ9AppL0JbumVAiB1pYju8pKu-P8IfgcOm19SLV4dJ4m2FS1ckD0A31U6tW5_2ZA/s320/CombiningEvidence.png" width="320" /></a></div><br /><div><div>where <i>A</i> is a set of possibilities, <i>E</i> is a set of sources of evidence, and <i>i<sub>e</sub></i> is the event that a source of evidence indicated whichever possibility it indicated (as opposed to indicating some other possibility). It's really a general formula for deriving probability from any number of conflicting, corroborating, and possibly biased sources of evidence. So long as the sources of evidence are independent and they each indicate exactly one of a limited number of possibilities, this formula applies. For instance, suppose you have several people making claims about something that happened, some agreeing and some disagreeing. How do you weigh their testimony against each other? If you can find a way to estimate how likely each person was to make the claim that he or she claimed given each of the possibilities, this formula provides you with a way forward.</div><div><div><br /></div></div><div>Here's a math paper that states the formula as a theorem and proves it: <a href="https://arxiv.org/abs/2003.03970" target="_blank">Bayes' Theorem Under Conditional Independence</a>. It's Theorem 4. (The paper expresses the formula with different symbols, but it's essentially the same thing.) I haven't found it anywhere else in print or online. Since the numerator on the right-hand side is the basis for <a href="https://en.wikipedia.org/wiki/Naive_Bayes_classifier" target="_blank">naive Bayes classifiers</a>, maybe it's included in some AI course material somewhere. I suppose it falls between two stools; it isn't one of the most basic applications of Bayes' theorem, so it wouldn't be in introductory material, but to a professional mathematician, it might seem too obvious to deserve mentioning.</div><div><div><br /></div></div></div>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-30750614298443290172023-02-04T07:51:00.023-05:002023-03-06T13:40:55.195-05:00Professor Kirke's House<p>The filming location for Professor Kirke's house in the BBC version of "The Lion, the Witch and the Wardrobe" (<a href="https://www.imdb.com/title/tt0094500/" target="_blank">IMDb</a>) is <a href="https://en.wikipedia.org/wiki/Llanvihangel_Court">Llanvihangel Court</a>. Some of the rooms, especially the spare room that contains the wardrobe, look like they might be sets, but everything else appears to be filmed there. For evidence, watch <a href="https://www.youtube.com/watch?v=YxwdwSdnEeU&t=2943s" style="font-family: inherit;">this YouTube copy</a><span style="font-family: inherit;"> of "The Lion, the Witch and the Wardrobe", specifically at these times, noting:</span></p><p></p><ul style="text-align: left;"><li><span style="font-family: inherit;">3:16 - the front of the house,</span></li><li><span style="font-family: inherit;">23:10 - the fountain,</span></li><li><span style="font-family: inherit;">48:31 - the garden house (guardhouse) next to a red tree and</span></li><li><span style="font-family: inherit;">49:01 - the stained-glass window,</span></li></ul><p></p><p><span style="font-family: inherit;">and compare them to these <a data-saferedirecturl="https://www.google.com/url?q=http://www.ewyaslacy.org.uk/Ewyas-Lacy-environs-/Digital-Archive-Collection-of-Photographs-of-Llanvihangel-Court/2017/rs_ewy_0337&source=gmail&ust=1673615415126000&usg=AOvVaw2H0iMoMACTpWXVZ1-SAi8I" href="http://www.ewyaslacy.org.uk/Ewyas-Lacy-environs-/Digital-Archive-Collection-of-Photographs-of-Llanvihangel-Court/2017/rs_ewy_0337" target="_blank">2017 photographs of Llanvihangel Court</a> and the photo of the stained-glass window here: <a data-saferedirecturl="https://www.google.com/url?q=https://stainedglass.llgc.org.uk/object/4868&source=gmail&ust=1673615415125000&usg=AOvVaw2OizSL-bjl-rGryddkwn57" href="https://stainedglass.llgc.org.uk/object/4868" target="_blank">Elizabeth I from Elizabeth I and the Family of Charles I - work from Stained Glass in Wales (llgc.org.uk)</a></span></p><p>Unlike some other locations for the series, the house wasn't mentioned in closing credits. At the time of filming, it was a private house and the owners may have not wanted publicity. In recent years, the house appears to have hosted tours and events. It has a web site: <a href="https://www.llanvihangelcourt.com/">https://www.llanvihangelcourt.com/</a></p><p>I found the house by looking for recognizable objects in scenes set inside and outside the house and searching the Internet for them. It took me a while to notice the stained-glass window, but eventually I did, and that was the key. It depicts a lady that looks like Queen Elizabeth I. On closer inspection, I saw that there is a gentleman on either side of the lady and one of them is laying down a cloak. I recalled there was an incident where Sir Walter Raleigh laid down his cloak for Queen Elizabeth so she would not have to step in a puddle. Googling "elizabeth", "cloak", "raleigh", and "stained glass" as required terms turned up the "Stained Glass in Wales" web page mentioned above, which identifies the location of the stained-glass window: Llanvihangel Court. </p><p>As for the puddle incident, Wikipedia says it is probably apocryphal. Another source flatly states that it never happened because the first known written reference to it was written 80 years later by a Thomas Fuller, who sometimes fabricated stories. Of course, it is a <i>non sequitur</i> to go from "this person sometimes fabricates stories" and "this person was the first person we know of to write about it" to "we know this never happened", so I disagree. You can read Thomas Fuller' account of the incident <a href="https://books.google.com/books?id=XvYm-o5Gw_UC&pg=162#v=onepage&q&f=false" target="_blank">here</a>. Actually, it isn't so much an account as a mention. Quoting Fuller: "his introduction into court is said to have born an elder date, from the time he spread his plush cloak for Her Majesty to step upon over a wet place". It seems like Fuller is stating hearsay or repeating tradition. As it is written, I don't know why he would have made it up himself. For myself, I would rate the incident to be about as likely to be true as any bit of remembered information about a historical person, or any other unproven statement made in the book it was included in.</p>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-2337047407839799222021-09-15T15:51:00.000-04:002021-09-15T15:51:06.787-04:00"New Symbols for Base-16 and Base-256 Numerals" Bested on Reddit<p>In 2017, I co-authored a paper that presented a system of hexadecimal digits that could be combined as ligatures to form base-256 digits. It's online <a href="http://www.ijcset.com/docs/IJCSET17-08-06-027.pdf" target="_blank">here</a>. In 2019, someone posted a system of base-256 digits on Reddit that I and the other author agree is better: <a href="https://www.reddit.com/r/conlangs/comments/dcln4t/made_a_surprisingly_simple_base256_numerical/">Made a surprisingly simple base-256 numerical system...</a></p><p>In our paper, we enumerated 9 desirable qualities for a set of hexadecimal digits in order to evaluate different sets. This other system has/hasn't the qualities as follows:</p><p></p><ul style="text-align: left;"><li>MNE: Yes</li><li>STR: No</li><li>LIG: Yes</li><li>AMB: Yes</li><li>DSP: Yes</li><li>BIN: Yes</li><li>0: No</li><li>1: No</li><li>TRN: Yes</li></ul><div>This system has 6 of the 9 enumerated qualities. This is 2 fewer qualities than our published system, but I nevertheless think it is a better system because 1) its base-256 "ligatures" are easier to write and 2) I can remember how these digits encode binary numbers much more easily than our published system, and therefore this system would be easier for me to use. Well done, <a href="https://www.reddit.com/user/Tuckertcs/">u/Tuckertcs</a>, whoever you are.</div><div><br /></div><div>On another topic, in our paper we suggested the name "sedecisedecimal" for base-256 numbers. I still think it's the best Latin/Greek-based name for them. Would anyone seriously prefer "<a href="https://en.wikipedia.org/wiki/List_of_numeral_systems#Standard_positional_numeral_systems" target="_blank">duocentehexaquinquagesimal</a>"?</div><p></p>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-52307323632785375142019-11-19T19:11:00.001-05:002019-11-19T19:16:33.924-05:00Bill Watterson's Political Cartoons<br />
<div>While attending Kenyon College, Bill Watterson (the author of Calvin and Hobbes) was editorial cartoonist for the school newspaper, The Kenyon Collegian. Most weeks during his tenure, they printed a cartoon of his, usually a political cartoon. Since the Kenyon Collegian's <a href="https://digital.kenyon.edu/collegian/" target="_blank">archives are available online</a>, you can see them all. The following is a complete set of links to issues of the Kenyon Collegian that have these cartoons: </div><ul><li>Volume CIV (1976-1977), issues <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1942&context=collegian&#page=3" target="_blank">2</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1943&context=collegian&#page=5" target="_blank">3</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1944&context=collegian&#page=3" target="_blank">4</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1945&context=collegian&#page=3" target="_blank">5</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1946&context=collegian&#page=3" target="_blank">6</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1947&context=collegian&#page=3" target="_blank">7</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1948&context=collegian&#page=3" target="_blank">8</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1949&context=collegian&#page=2" target="_blank">9</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1950&context=collegian&#page=3" target="_blank">10</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1951&context=collegian&#page=3" target="_blank">11</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1952&context=collegian&#page=3" target="_blank">12</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1953&context=collegian&#page=3" target="_blank">13</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1954&context=collegian&#page=3" target="_blank">14</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1955&context=collegian&#page=3" target="_blank">15</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1956&context=collegian&#page=3" target="_blank">16</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1957&context=collegian&#page=3" target="_blank">17</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1958&context=collegian&#page=3" target="_blank">18</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1959&context=collegian&#page=3" target="_blank">19</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1960&context=collegian&#page=3" target="_blank">20</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1961&context=collegian&#page=3" target="_blank">21</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1962&context=collegian&#page=3" target="_blank">22</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1963&context=collegian&#page=3" target="_blank">23</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1964&context=collegian&#page=3" target="_blank">24</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1965&context=collegian&#page=3" target="_blank">25</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1966&context=collegian&#page=3" target="_blank">26</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1967&context=collegian&#page=3" target="_blank">27</a></li>
<li>Volume CV (1977-1978), issues <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1969&context=collegian&#page=3" target="_blank">1</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1970&context=collegian&#page=3" target="_blank">2</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1971&context=collegian&#page=3" target="_blank">3</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1972&context=collegian&#page=3" target="_blank">4</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1973&context=collegian&#page=3" target="_blank">5</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1974&context=collegian&#page=3" target="_blank">6</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1975&context=collegian&#page=3" target="_blank">7</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1976&context=collegian&#page=3" target="_blank">8</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1977&context=collegian&#page=3" target="_blank">9</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1978&context=collegian&#page=3" target="_blank">10</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1979&context=collegian&#page=3" target="_blank">11</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1980&context=collegian&#page=3" target="_blank">12</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1981&context=collegian&#page=3" target="_blank">13</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1982&context=collegian&#page=3" target="_blank">14</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1983&context=collegian&#page=3" target="_blank">15</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1984&context=collegian&#page=3" target="_blank">16</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1985&context=collegian&#page=3" target="_blank">17</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1986&context=collegian&#page=3" target="_blank">18</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1987&context=collegian&#page=3" target="_blank">19</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1988&context=collegian&#page=3" target="_blank">20</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1989&context=collegian&#page=2" target="_blank">21</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1990&context=collegian&#page=3" target="_blank">22</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1991&context=collegian&#page=3" target="_blank">23</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1992&context=collegian&#page=3" target="_blank">24</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1993&context=collegian&#page=3" target="_blank">25</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1994&context=collegian&#page=3" target="_blank">26</a></li>
<li>Volume CVI (1978-1979), issues <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1996&context=collegian&#page=3" target="_blank">1</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1997&context=collegian&#page=3" target="_blank">2</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1998&context=collegian&#page=3" target="_blank">3</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=1999&context=collegian&#page=3" target="_blank">4</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2000&context=collegian&#page=3" target="_blank">5</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2001&context=collegian&#page=3" target="_blank">6</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2002&context=collegian&#page=3" target="_blank">7</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2003&context=collegian&#page=3" target="_blank">8</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2004&context=collegian&#page=3" target="_blank">9</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2005&context=collegian&#page=3" target="_blank">10</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2006&context=collegian&#page=3" target="_blank">11</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2007&context=collegian&#page=3" target="_blank">12</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2008&context=collegian&#page=4" target="_blank">13</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2009&context=collegian&#page=3" target="_blank">14</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2010&context=collegian&#page=3" target="_blank">15</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2011&context=collegian&#page=3" target="_blank">16</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2012&context=collegian&#page=3" target="_blank">17</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2013&context=collegian&#page=3" target="_blank">18</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2014&context=collegian&#page=3" target="_blank">19</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2015&context=collegian&#page=3" target="_blank">20</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2016&context=collegian&#page=3" target="_blank">21</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2017&context=collegian&#page=3" target="_blank">22</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2018&context=collegian&#page=3" target="_blank">23</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2019&context=collegian&#page=3" target="_blank">24</a></li>
<li>Volume CVII (1979-1980), issues <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2020&context=collegian&#page=3" target="_blank">1</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2021&context=collegian&#page=3" target="_blank">2</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2022&context=collegian&#page=3" target="_blank">3</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2023&context=collegian&#page=3" target="_blank">4</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2024&context=collegian&#page=3" target="_blank">5</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2025&context=collegian&#page=3" target="_blank">6</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2026&context=collegian&#page=3" target="_blank">7</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2027&context=collegian&#page=3" target="_blank">8</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2028&context=collegian&#page=3" target="_blank">9</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2029&context=collegian&#page=3" target="_blank">10</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2030&context=collegian&#page=3" target="_blank">11</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2031&context=collegian&#page=3" target="_blank">12</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2032&context=collegian&#page=3" target="_blank">13</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2033&context=collegian&#page=3" target="_blank">14</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2034&context=collegian&#page=4" target="_blank">15</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2035&context=collegian&#page=3" target="_blank">16</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2036&context=collegian&#page=3" target="_blank">17</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2037&context=collegian&#page=3" target="_blank">18</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2038&context=collegian&#page=3" target="_blank">19</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2039&context=collegian&#page=3" target="_blank">20</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2040&context=collegian&#page=3" target="_blank">21</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2041&context=collegian&#page=3" target="_blank">22</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2042&context=collegian&#page=3" target="_blank">23</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2043&context=collegian&#page=3" target="_blank">24</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2044&context=collegian&#page=3" target="_blank">25</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2045&context=collegian&#page=3" target="_blank">26</a>, <a href="https://digital.kenyon.edu/cgi/viewcontent.cgi?article=2046&context=collegian&#page=3" target="_blank">27</a></li>
</ul>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-42565790996691104252019-11-19T19:04:00.001-05:002019-11-19T19:04:44.300-05:00It's Rational to Believe in JesusSo says Aron Wall (and me). He has posted a fair amount of material on his web site that explains how believing what the Bible says about Jesus can be rational. Several of those postings amount to as good a case for Christianity as I've seen expressed anywhere, so I'm posting links to them for your benefit (and my own; I will re-read them). They are listed in an order that I think makes sense for them to be read if they were to be read in sequence.<br />
<div>
<br />
<hr />
<br /></div>
<div>
There is good evidence that Jesus was resurrected from the dead and that he is the Messiah:</div>
<ol>
<li><a href="http://wall.org/~aron/evidence.htm" target="_blank">Can Religion Be Based on Evidence?</a></li>
<li><a href="http://www.wall.org/~aron/blog/theology-less-speculative-than-quantum-gravity/" target="_blank">Theology: Less Speculative than Quantum Gravity</a></li>
<li><a href="http://www.wall.org/~aron/blog/christianity-is-based-on-observations/" target="_blank">Christianity Is Based on Observations</a></li>
<li><a href="http://www.wall.org/~aron/blog/let-us-calculate/" target="_blank">Let Us Calculate</a></li>
<li><a href="http://www.wall.org/~aron/blog/christianity-is-true/" target="_blank">Christianity Is True</a></li>
</ol>
<div>
The Old Testament part of the Bible is not a fabrication:</div>
<div>
<ul>
<li><a href="http://www.wall.org/~aron/blog/some-comments-on-biblical-history/" target="_blank">Some Comments on Biblical History</a></li>
</ul>
<div>
Christianity is more likely to be true than other religions:</div>
</div>
<div>
<ol>
<li><a href="http://www.wall.org/~aron/blog/comparing-religions-i-introduction/" target="_blank">Introduction</a></li>
<li><a href="http://www.wall.org/~aron/blog/comparing-religions-ii-world-evangelism/" target="_blank">World Evangelism</a></li>
<li><a href="http://www.wall.org/~aron/blog/comparing-religions-iii-ancient-roots/" target="_blank">Ancient Roots</a></li>
<li><a href="http://www.wall.org/~aron/blog/comparing-religions-iv-supernatural-claims/" target="_blank">Supernatural Claims</a></li>
<li><a href="http://www.wall.org/~aron/blog/comparing-religions-v-historical-accounts/" target="_blank">Historical Accounts</a></li>
<li><a href="http://www.wall.org/~aron/blog/comparing-religions-vi-early-sources/" target="_blank">Early Sources</a></li>
</ol>
<br />
<hr />
<div>
<i>Note: I don't know Aron Wall personally; I've just read his blog and agree with his approach to Christian apologetics.</i></div>
</div>
mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-53933938977797872152019-04-27T08:09:00.003-04:002021-06-19T09:13:18.735-04:00How to Transfer User TonePrints from One Windows PC to AnotherThis post is for anyone who has <a href="https://www.tcelectronic.com/toneprints" target="_blank">TonePrints</a> for a <a href="https://www.tcelectronic.com/" target="_blank">TC Electronic</a> effect pedal on one computer and wants to transfer them to another computer.<br />
<br />Version 4.1.06 of the TonePrint application stores user TonePrints in a file named "database.sqlite3" in the user's app data folder, which, in Windows 10, is a hidden folder named "C:\Users\[user name]\AppData\Roaming". If you want to replace your user TonePrints on one computer with user TonePrints from another computer, just copy this file from the one computer to the other.<br />
<br />If you want to copy TonePrints from one PC to another, but you want to keep your old TonePrints, too, simply copying the file won't work, because it will overwrite TonePrints. For that purpose, I wrote some scripts for Windows PC that will grab TonePrints from one PC and install them on another PC without deleting the TonePrints that are already there. You can download them in a zip file here:<br />
<ul>
<li><a href="http://somerby.net/mack/TonePrint_transfer.zip">TonePrint_transfer.zip</a></li>
</ul>
To use them, download the zip file, extract the contents to an empty folder, then follow the instructions in the file named "README.txt" that is included in the zip file. They should work with version 4.1.06 of the PC TonePrint application.<br />
<br />I don't know how to transfer user TonePrints from a smartphone to a PC, or vice versa. The smartphone app might have a database.sqlite3, but I have not been able to locate it on my phone.<div><br /></div><div>UPDATE: the scripts won't work for new versions of the TonePrint application (e.g. 4.4.03). The database file name apparently changed to "databaseV1.sqlite3" or something.</div>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com1tag:blogger.com,1999:blog-35719397.post-51859980512517389842019-04-20T09:31:00.000-04:002019-06-16T19:34:27.266-04:00How to Build & Run Vampire ATP in CygwinVampire is an Automated Theorem Prover. It was apparently developed on Linux, but it can be run on Windows using Cygwin, which provides the build tools, software packages and operating system services that Vampire depends on. Here are some instructions for building and running Vampire in 64-bit Windows 10. A note on building Vampire with Cygwin can be found on Vampire's GitHub page, <a href="https://github.com/vprover/vampire/wiki/Vampire-under-Cygwin" target="_blank">here</a>.<br />
<h4>
Install Cygwin:</h4>
<ol>
<li>Install <a href="https://cygwin.com/install.html" target="_blank">Cygwin</a> and its default packages.</li>
<li>Install these Cygwin packages:<br />
<ul>
<li>cygrunsrv</li>
<li>gcc-core</li>
<li>gcc-g++</li>
<li>make </li>
<li>zlib</li>
<li>zlib-devel</li>
<li>zlib-devel</li>
<li>zlib0</li>
</ul>
</li>
<li>Move C:\cygwin64\bin to the front of the %PATH% environment variable.</li>
</ol>
<h4>
Install Git Command-Line Tools</h4>
<ol>
<li>Install the Git command-line tools. An installer can be downloaded from <a href="https://git-scm.com/download/win">https://git-scm.com/download/win</a>.</li>
</ol>
<h4>
Install the Z3 Theorem Prover</h4>
<ol>
<li>Download a Win64 release of Z3 from <a href="https://github.com/Z3Prover/z3/releases">https://github.com/Z3Prover/z3/releases</a>.</li>
<li>Make z3.dll available from PATH, either by putting z3.dll from this release in some folder that is in PATH, or by adding the Z3 "bin" folder to PATH.</li>
</ol>
<h4>
Build Vampire</h4>
<ol>
<li>Pull Vampire's Git repository from GitHub (<a href="https://github.com/vprover/vampire">https://github.com/vprover/vampire</a>). </li>
<li>In Vampire's Makefile, add "-D_GNU_SOURCE" to CXXFLAGS and CCFLAGS.</li>
<li>Change all instances of "<span style="font-family: "courier new" , "courier" , monospace;">exit(0);</span>" in the source code to "<span style="font-family: "courier new" , "courier" , monospace;">_Exit(0);</span>". This is needed to prevent the child processes forked by the main Vampire process from calling the main process' atexit handlers.</li>
<li>Create a folder named "include" in the Vampire repository's main folder.</li>
<li>Copy the contents of the Z3 "bin" folder to that "include" folder.</li>
<li>Compile and link vampire by issuing the command "make vampire_z3_rel" from a command prompt.</li>
</ol>
<h4>
Start the Cygwin Windows Service</h4>
<ol>
<li>Start Cygwin's bash shell.</li>
<li>From the bash command prompt, issue the command "cygserver-config" to configure the Cygwin Windows service.</li>
<li>In Windows, right-click "This PC", select "Manage", then select "Services and Applications, then select "Services", then start the "CYGWIN cygserver" service.</li>
</ol>
<h4>
Run Vampire</h4>
<ol>
<li>Run Vampire from a command prompt. For example:<br />
<code>vampire_z3_rel_master_4055 --mode casc -t 300 "Problems\SET\SET159-6.p"</code></li>
<li>Vampire may not succeed in deleting the semaphores that it creates in the Cygwin server. To delete them, issue the command "<code>ipcs -s | grep '^s\\s' | cut -d' ' -f3|xargs -n 1 ipcrm -s</code>" from a command prompt. Alternatively, add a call to <code>system()</code> at the end of Vampire's <span style="font-family: "Courier New", Courier, monospace;">main</span> function that issues this command.</li>
</ol>
<ol></ol>
<ul></ul>
<ol></ol>
mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-12966321476447040622018-05-06T15:13:00.002-04:002018-05-06T15:13:34.797-04:00Historical and Cultural Context for the New TestamentYears ago, I read an article about how some Japanese toilets have special built-in features, including electric seat warmers. I thought this was weird. I would not want to sit on a warmed toilet seat and I didn't understand why someon else would. I couldn't explain it further than thinking to myself that different people feel differently about different things. Many years later, I talked to friend who had returned to the U.S. after spending some months living in Japan. He told me that the Japanese did not heat their entire homes. In cold seasons, they would heat the space they were staying in, or just their beds at night. After learning that, I was able figure out for myself why a Japanese person would want a seat warmer on their toilet. In an American, climate-controlled house, a room-temperature toilet seat is about 70 degrees Fahrenheit. A warmed toilet seat would be unpleasantly warm; to me it would feel like someone had just been sitting on the seat. Though I fully accept that other people use the same toilet seats as me, I don't like to be reminded of the fact. Now, what if room temperature is 50 degrees? Suddenly, the toilet seat warmer made sense to me.<br />
<br />
So what does this have to do with the Bible? It shows that context matters. If you want to understand why people in other cultures do what they do, it helps a lot to understand their circumstances. This goes doubly when people aren't just separated from you by distance and borders, but also by long stretches of time. I like to think the Bible mostly speaks for itself, but really there's a lot in it that might not make sense to a modern reader. In recent years, I've got into reading about the historical and cultural context of the Bible, expecially the New Testament. It's been so helpful to me in making sense of difficult parts that I think everyone who seriously wants to understand the Bible should do it. The following are some books that I recommend to anyone who wants to learn about the historical and cultural context of the New Testament. None are very long and I don't think any require much background apart from the Bible itself and some general knowledge of the Roman empire.<br />
<h4>
<a href="http://www.amazon.com/Poet-Peasant-Through-Eyes-Literary-Cultural/dp/0802819478/ref=sr_1_1?ie=UTF8&qid=1450889154&sr=8-1&keywords=poet+%26+peasant+and+through+peasant+eyes" target="_blank">Poet & Peasant and Through Peasant Eyes: A Literary-Cultural Approach to the Parables in Luke </a> by Kenneth E. Bailey </h4>
You may have heard in a sermon that it's important that the father of the Prodigal Son <i>runs</i>
to his son when he returns because in traditional Middle Eastern cultures, it's undignified for a grown man to run. Ok, maybe not, but in my experience those kinds of facts appear in sermons every once in a while. If you had, you might wonder where that bit of information from came from. The answer is the late <a href="https://shenango.org/bailey/" target="_blank">Kenneth E. Bailey</a>. Bailey wrote several books that interpret parts of the Bible using his knowledge of Middle Eastern culture and customs as
well as his knowledge of ancient Middle Eastern Bible translators and commentators. <i>Poet & Peasant and Through Peasant Eyes</i> are two of the earliest of these books, combined into one volume. In them, he explains how Jesus' contemporaries
would likely have understood several of Jesus' parables. <i>Not only that,</i>
he analyzes
the structure of the parables, showing that their sentences are
arranged deliberately in known rhetorical forms. I love these books and consequently I love Kenneth
Bailey. There is so much insight to be had from them. <br />
<h4>
<a href="https://www.amazon.com/Jerusalem-Time-Jesus-Joachim-Jeremias/dp/0800611365" target="_blank">Jerusalem in the Time of Jesus</a> by Joachim Jeremias</h4>
<h4>
<span style="font-weight: normal;">A wide-ranging book about... Jerusalem in the time of Jesus. Lots of information about the size, structure and economy of Jerusalem and other interesting things like trades and Jewish marriage customs. Perhaps most useful for understanding the New Testament are the sections about Jewish religious groups. They explain who the Pharisees and the Sadducees were, and who those "scribes" were that Jesus was arguing with in the Gospels.</span></h4>
<h4>
<a href="http://www.amazon.com/Josephus-Thrones-Blood-Flavius/dp/1557482667/ref=sr_1_1?ie=UTF8&qid=1450888429&sr=8-1&keywords=josephus+thrones+of+blood" target="_blank">Josephus: Thrones of Blood</a></h4>
This is an abridgement and paraphrase of two of Josephus' books: <i>The Antiquities of the Jews</i> and <i>The Jewish War</i>. In these books, Josephus writes for a Roman audience, giving an account of the history of the Jewish nation from the times of the Herods to the destruction of Jerusalem in 70 <i>A.D</i>. The modern language of this paraphrase makes it easier to read than the older English translations, and it's not a long book. It's good for understanding the political climate that Jesus and his disciples lived in<i>.</i><br />
<h4>
<a href="http://www.amazon.com/Slaves-Citizens-Sons-Metaphors-Epistles/dp/0310451914/ref=sr_1_1?ie=UTF8&qid=1450888722&sr=8-1&keywords=slaves%2C+citizens%2C+sons" target="_blank">Slaves, Citizens, Sons: Legal Metaphors in the Epistles</a> by Francis Lyall</h4>
<h4>
<span style="font-weight: normal;">In
this book, Lyall explains Roman, Greek and Hebrew law pertaining to
slavery, citizenship, sonship, inheritance and adoption. Dry stuff?
No, because it's useful for understanding what Paul means when he writes
about being adopted into God's family, being "sons of God", being a
"bondservant of Christ", etc. It's also interesting to see how some
things that so many modern people take for granted, like the fact that
laws apply to all individuals in a society, just weren't so in the
ancient world, especially Rome. In Roman law, only free men (usually
heads of households) were subject to law. Everyone else -- women,
children, slaves and unemancipated sons -- were considered to be
property and therefore considered to be "objects" as far as laws were concerned. </span></h4>
<h4>
<a href="http://www.biblestudytools.com/apocrypha/" target="_blank">The Old Testament Apocrypha</a></h4>
For most of my life, I never read the Apocrypha or thought much about them. But then I read N. T. Wright's <span class="mw-headline" id=".22Christian_Origins_and_the_Question_of_God.22_series">series, "Christian Origins and the Question of God". In these books, he references them so often that I had to read them to know what he was talking about. Most were boring to me. 2 Esdras is strange. I found Ben Sirach's proverbs to be interesting; some are good, but others are worldly and self-interested. Some are even contrary to what Jesus taught, which reinforces my Protestant belief that it should be canonical. Even so, these books were read by Jesus' contemporaries and are likely to represent their worldview. Also, they fill in some of the gap between Old Testament and New Testament history.</span>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-80808851452720917682018-03-31T20:58:00.000-04:002018-03-31T20:58:00.906-04:00Improvements to somerby.net/mack/logic After a spate of good input from commenters, I've improved <a href="http://somerby.net/mack/logic">somerby.net/mack/logic</a> in several ways:<br />
<ul>
<li>Instead of showing just one counterexample, clicking the "Counterexample" button more than once causes the application to cycle through different counterexamples for the current statement.</li>
<li>There is a new button: the "Example" button, which is the opposite of the "Counterexample" button. </li>
<li>Propositions (nullary predicates) can now be lower-case letters as well as upper-case letters.</li>
<li>There is now a "therefore" operator -- ',' -- apostrophe comma apostrophe, for representing logical arguments.</li>
</ul>
mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-28272823424716435902017-10-15T22:44:00.000-04:002019-05-27T22:47:38.380-04:00The Video Fuzz: a Fuzz Pedal Based on the S9014 Transistor<br />
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I built a fuzz pedal for my guitar using parts from an old, off-brand Nintendo accessory called the "Video Shooter" and a broken Fender Frontman amplifier. The pedal is based on the <a href="http://home-wrecker.com/bazz.html" target="_blank">Bazz Fuss</a>, which is a simple fuzz circuit that is easy to modify. I wanted to use parts from the Nintendo accessory as much as possible, so I designed the circuit around the accessory's enclosure and the S9014 transistors I found inside. One S9014 did not put out enough gain to work as a guitar effect, so, after reading <a href="https://groupdiy.com/index.php?topic=62317.msg789726#msg789726" target="_blank">this forum thread</a>, I joined two transistors into what is called a "Darlington transistor". The happy result was sufficient gain and a very well-behaved fuzz pedal. Unlike some other Bazz Fuss derivatives I have encountered, it doesn't fizzle or sputter when connected to a guitar with high-impedance pickups and or when I roll off the volume on the guitar. Its gain is moderate. <!--Chords sound fuzzy. Single notes played high on the neck sound creamy. Single notes played low on the neck sound like "Spirit in the Sky". Notes in between sound buzzy like a bagpipe chanter. --> I call it the "Video Fuzz" in honor of the <a href="http://nostalbit.com/placo-video-shooter-for-nes_863.html" target="_blank">NES accessory</a>. </div>
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Since it's so well-behaved, I'm sharing the schematic here in case anyone wants a good starting point for their own fuzzy experimentations:<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiesXaivTYHLI3cMkB9Cis6bNnP37orw2DXYdgVc-exDv_Npvr-uIrKtzR1nWuyq58sCwfzKH1OoqxSLCo6zLM_uDisKrk_a2blStGny9nhpzSODeZby0Ypr_bpEjH61AVP0xEi/s1600/VideoShooterFuzzSchematic.png" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="475" data-original-width="515" height="369" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiesXaivTYHLI3cMkB9Cis6bNnP37orw2DXYdgVc-exDv_Npvr-uIrKtzR1nWuyq58sCwfzKH1OoqxSLCo6zLM_uDisKrk_a2blStGny9nhpzSODeZby0Ypr_bpEjH61AVP0xEi/s400/VideoShooterFuzzSchematic.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Video Shooter Fuzz Schematic</td></tr>
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If you compare the circuit to the original Bazz Fuss, you'll notice I added a resistor in front of the pot going to the output. I did this because the output is well above unity gain even with 47k Ohms resistance; I'd never want to use the full range of a 100k Ohm pot. I added a 2.2k Ohm resistor at at the collector of the transistor to set the fuzz to a level that sounded good to me. I could have put a pot at the collector to have a two-knob fuzz, but instead I opted to make a one-knob fuzz and control the fuzz level with my guitar's volume knob. For the switch and jack wiring, I used the wiring shown second to last in <a href="http://diystrat.blogspot.com/2009/06/how-to-wire-up-stomp-box-effects-pedal.html" target="_blank">DIYStrat: Wiring a Stomp Box</a>. Here is the finished pedal:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5konat3Bd5eSVkq6VTwtQ2kmb6gojV0mpgMy7gzrdgkwPUHRJANyxpBrvdUk-jQIkEair__k0F75doGgPIxikU6UXrnb4ohhaLryPIXMvxO7rwbfaLoCNxzYmb4EmrWMM3IQo/s1600/20170926_210007.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5konat3Bd5eSVkq6VTwtQ2kmb6gojV0mpgMy7gzrdgkwPUHRJANyxpBrvdUk-jQIkEair__k0F75doGgPIxikU6UXrnb4ohhaLryPIXMvxO7rwbfaLoCNxzYmb4EmrWMM3IQo/s320/20170926_210007.jpg" width="180" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzZle0Jc7o-LkhrKqWY8lCVMDTMNdPn5GAUXaZApHVVUuK_QzP3zLwaIGOmkSfaKQLNje4MXgG2O3hysejYaJGi6H3OysNv9wah9HsOFIT0OXb6Iv_qfNlOkGiDbUBGGKbVq3W/s1600/20170926_205851.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzZle0Jc7o-LkhrKqWY8lCVMDTMNdPn5GAUXaZApHVVUuK_QzP3zLwaIGOmkSfaKQLNje4MXgG2O3hysejYaJGi6H3OysNv9wah9HsOFIT0OXb6Iv_qfNlOkGiDbUBGGKbVq3W/s320/20170926_205851.jpg" width="180" /></a></div>
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This is the fuzz circuit by itself, minus the 50k Ohm pot:<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaXLynzF_LDmnT_3qYnD-Fi4uCjmHxOAbc3TRmqrjztCr2orVo52YR7Iwc8FRpTX0_MNxcDs6aFd3BHzGRgLsjdj3h-rSDhc1Q4pr_-dKWI_USjMGxg6t81Vrylr2ezdk_DBoh/s1600/20170924_123118.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="900" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaXLynzF_LDmnT_3qYnD-Fi4uCjmHxOAbc3TRmqrjztCr2orVo52YR7Iwc8FRpTX0_MNxcDs6aFd3BHzGRgLsjdj3h-rSDhc1Q4pr_-dKWI_USjMGxg6t81Vrylr2ezdk_DBoh/s320/20170924_123118.jpg" width="180" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Hand-Soldered, Point-to-Point Wiring!</td></tr>
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I won't show you rest of the insides; you've seen enough of the sausage being made.<br />
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And here are some sound clips: <a href="https://soundcloud.com/mackenzie-cumings/video-fuzz-sample-1" target="_blank">(1)</a>, <a href="https://soundcloud.com/mackenzie-cumings/video-fuzz-sample-2" target="_blank">(2)</a>. Interestingly, when I was first wiring a Bazz Fuss circuit, I connected the collector and the pot to the negative end of a battery but not to ground and the resulting circuit made no distortion and behaved like a buffer. <br />
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<br />mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com5tag:blogger.com,1999:blog-35719397.post-59263612969276142762017-09-04T11:42:00.003-04:002017-09-04T20:51:54.374-04:00Alternative Hexadecimal Digits: Published<a href="https://odo.lv/valdis_vitolins_cv" target="_blank">Valdis</a> and I wrote up our final version of the hexadecimal digits in the form of a proposal and got it published in the <a href="http://www.ijcset.com/" target="_blank">IJCSET</a>. See <a href="http://www.ijcset.com/docs/IJCSET17-08-06-027.pdf" target="_blank">here</a>. The proposal includes descriptions and assessments of various other proposed sets of hexadecimal digits. Unfortunately, we missed a good set called <a href="http://yawar.blogspot.com/2016/10/the-birkana-hexadecimal-number-symbols.html" target="_blank">Birkana</a> that are rune-like symbols. <br />
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In case anyone wants to see our digits used for practical purposes, I made a <a href="http://www.somerby.net/mack/clock.html" target="_blank">JavaScript digital clock</a> that uses our digits to display hours, minutes and seconds as hexadecimal numbers.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxY7cyHIkw0sQ2sI7p1hU82FRQnNgN1PdNqJ-gzWsIlK4ChBWgblPfm-rq2K7rjBT1SGqqt2RxUwCTWFzkWi5XWbZNaDAFOrE-8ZbniPYeGverkeLK7_LMjepy1x8G8ihghzFC/s1600/hex_clock.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="126" data-original-width="525" height="76" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxY7cyHIkw0sQ2sI7p1hU82FRQnNgN1PdNqJ-gzWsIlK4ChBWgblPfm-rq2K7rjBT1SGqqt2RxUwCTWFzkWi5XWbZNaDAFOrE-8ZbniPYeGverkeLK7_LMjepy1x8G8ihghzFC/s320/hex_clock.png" width="320" /></a></div><br />
mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-82928695548122082922016-04-17T17:55:00.002-04:002016-04-17T17:55:55.509-04:00A Cancel Button for somerby.net/mack/logicGood news: <a href="http://somerby.net/mack/logic">somerby.net/mack/logic</a> now has a cancel button. If a decision is taking too long, you can cancel it and continue working. You don't have to close the browser anymore.<br />
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Bad news: somerby.net/mack/logic is not working in Internet Explorer, or at least it isn't working in Internet Explorer 11 on my computer. I'll try to fix it soon.<br />
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It works for me in Chrome and Firefox. I'd appreciate it if anyone tells me whether or not its working for them in other browsers. Just leave a comment on this post.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com1tag:blogger.com,1999:blog-35719397.post-28758131189509753922016-02-15T21:21:00.004-05:002016-02-15T21:40:29.950-05:00A New Web App: "Expanding Quine's Definitions"I've been messing around with code that creates symbolic definitions of the number one for quite a while: <a href="http://mackwai.blogspot.com/search/label/the%20number%20one">these blog posts</a> show some of the results. All this messing around has culminated in a web app:<br />
<a href="http://somerby.net/mack/definitions/" target="_blank">http://somerby.net/mack/definitions/</a>. With it, you can expand Quine's definition of the number one interactively in various ways, and also many other of his definitions. What fun! mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-74912360219193980972015-12-29T17:35:00.000-05:002016-02-15T21:38:38.708-05:00The Number One, Part ThreeIn an earlier post, <a href="http://mackwai.blogspot.com/2007/05/what-it-means-to-be-number-one.html">"What it Means to be Number One"</a>, I presented Quine's definition of the number one, which is defined in terms of four basic mathematical constructs: class membership (<span id="class_membership">⌜α ϵ β⌝</span>), universal generalization (<span id="universal_generalization">⌜(α)ϕ⌝</span>), joint denial (<span id="joint_denial">⌜(ϕ ↓ ψ)⌝</span>), and class abstraction (<span id="class_abstraction">⌜<span class="over">α^</span>ϕ⌝</span>). It is six pages of dense text. In another post, <a href="http://mackwai.blogspot.com/2007/05/number-one-part-two.html">"The Number One, Part Two"</a>, I claimed that the length of the definition is "the result of giving the number one a precise, complete, logical
definition without resorting to using any numbers except zero". This is not quite true. I've since realized that the size comes not from precision, completeness nor logicality, but from Quine's definition of negation, which is<br />
<br />
⌜∼ϕ⌝ <i>for</i> ⌜(ϕ ↓ ϕ)⌝<br />
<br />
This is definition is important because it enables his system to define all truth-functional connectives in terms of just one truth-functional connectives, thereby minimizing the number of basic constructs in his system. But it does have the effect of making expanded definitions quite large. You see, any time a negation is expanded, the expression which is negated (ϕ) is duplicated in the resulting expansion. If that expression also contains negated expressions, then those expressions are quadrupled, and if those negated expressions contain negated expressions, they are octupled, and so on. The result is an exponential relationship between the size of an expanded definition and the number of layers of negation in the definition. Those of you who know computer science know that exponential relationships mean huge outputs for all but the smallest inputs. Hence the 6-page definition of one. If we expand all constructs contained in Quine's definition of one <i>except</i> negation, the result is not so big. It is<br />
<br />
<span class="over"><i>x</i>^</span>∼(<i>y</i>)∼(∼(<i>y</i> ϵ <i>x</i>) ↓ ∼(<span class="over">α^</span>(∼(α ϵ <i>x</i>) ↓ ∼(α ϵ <span class="over">α′^</span>(∼(α′ ϵ <span class="over">α′′^</span>((α′′′)(∼∼(∼(α′′′ ϵ α′′) ↓ <sub>▪</sub>α′′′ ϵ <i>y</i>) ↓ ∼∼(∼(α′′′ ϵ <i>y</i>) ↓ <sub>▪</sub>α′′′ ϵ α′′))))))) ϵ <span class="over">α^</span>((α′)(∼∼(∼(α′ ϵ α) ↓ <sub>▪</sub>α′ ϵ <span class="over"><i>x</i>′^</span>(∼((α′′)(∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′) ↓ ∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′))))) ↓ ∼∼(∼(α′ ϵ <span class="over"><i>x</i>′^</span>(∼((α′′)(∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′) ↓ ∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′))))) ↓ <sub>▪</sub>α′ ϵ α)))))<br />
<br />
which is not shockingly complicated. Even if we expand statements of membership in and of class abstractions in this definition (which is something I did not do in "What it means to be..."), the definition of one is still just<br />
<br />
<span class="over"><i>x</i>^</span>∼(<i>y</i>)∼(∼(<i>y</i> ϵ <i>x</i>) ↓ ∼(∼(γ)∼(∼(∼(β)∼(∼((α)(∼∼(∼(α ϵ β) ↓ <sub>▪</sub>∼(γ′)∼(∼(α ϵ γ′) ↓ ∼(α′)∼(∼(α′ ϵ γ′) ↓ (∼(α′ ϵ <i>x</i>) ↓ ∼(∼(γ′′)∼(∼(α′ ϵ γ′′) ↓ ∼(α′′)∼(∼(α′′ ϵ γ′′) ↓ <sub>▪</sub>∼(∼(γ′′′)∼(∼(α′′ ϵ γ′′′) ↓ ∼(α′′′)∼(∼(α′′′ ϵ γ′′′) ↓ <sub>▪</sub>(α′′′′)(∼∼(∼(α′′′′ ϵ α′′′) ↓ <sub>▪</sub>α′′′′ ϵ <i>y</i>) ↓ ∼∼(∼(α′′′′ ϵ <i>y</i>) ↓ <sub>▪</sub>α′′′′ ϵ α′′′)))))))))))) ↓ ∼∼(∼(∼(γ′)∼(∼(α ϵ γ′) ↓ ∼(α′)∼(∼(α′ ϵ γ′) ↓ (∼(α′ ϵ <i>x</i>) ↓ ∼(∼(γ′′)∼(∼(α′ ϵ γ′′) ↓ ∼(α′′)∼(∼(α′′ ϵ γ′′) ↓ <sub>▪</sub>∼(∼(γ′′′)∼(∼(α′′ ϵ γ′′′) ↓ ∼(α′′′)∼(∼(α′′′ ϵ γ′′′) ↓ <sub>▪</sub>(α′′′′)(∼∼(∼(α′′′′ ϵ α′′′) ↓ <sub>▪</sub>α′′′′ ϵ <i>y</i>) ↓ ∼∼(∼(α′′′′ ϵ <i>y</i>) ↓ <sub>▪</sub>α′′′′ ϵ α′′′)))))))))))) ↓ <sub>▪</sub>α ϵ β))) ↓ ∼(β ϵ γ))) ↓ ∼(α)∼(∼(α ϵ γ) ↓ <sub>▪</sub>(α′)(∼∼(∼(α′ ϵ α) ↓ <sub>▪</sub>∼(γ′)∼(∼(α′ ϵ γ′) ↓ ∼(<i>x</i>′)∼(∼(<i>x</i>′ ϵ γ′) ↓ <sub>▪</sub>∼((α′′)(∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′) ↓ ∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′)))))) ↓ ∼∼(∼(∼(γ′)∼(∼(α′ ϵ γ′) ↓ ∼(<i>x</i>′)∼(∼(<i>x</i>′ ϵ γ′) ↓ <sub>▪</sub>∼((α′′)(∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′) ↓ ∼∼(∼(α′′ ϵ <i>x</i>′) ↓ <sub>▪</sub>α′′ ϵ <i>x</i>′)))))) ↓ <sub>▪</sub>α′ ϵ α))))))<br />
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which is more than anyone would want to try to write or memory, but still not embarrassingly long. If we don't expand any of the truth-functional connectives, nor existential quantification, the result is something that it almost readable by a person who is familiar with symbolic logic:<br />
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<span class="over"><i>x</i>^</span>(∃<i>y</i>)(<i>y</i> ϵ <i>x</i> <sub>▪</sub> (∃γ)((∃β)((α)(α ϵ β<sub>▪</sub> ≡ <sub>▪</sub>(∃γ′)(α ϵ γ′ <sub>▪</sub> (α′)(α′ ϵ γ′<sub>▪</sub> ⊃ (α′ ϵ <i>x</i> <sub>▪</sub> (∃γ′′)(α′ ϵ γ′′ <sub>▪</sub> (α′′)(α′′ ϵ γ′′<sub>▪</sub> ⊃ <sub>▪</sub>∼((∃γ′′′)(α′′ ϵ γ′′′ <sub>▪</sub> (α′′′)(α′′′ ϵ γ′′′<sub>▪</sub> ⊃ <sub>▪</sub>(α′′′′)(α′′′′ ϵ α′′′<sub>▪</sub> ≡ <sub>▪</sub>α′′′′ ϵ <i>y</i>)))))))))) <sub>▪</sub> β ϵ γ) <sub>▪</sub> (α)(α ϵ γ<sub>▪</sub> ⊃ <sub>▪</sub>(α′)(α′ ϵ α<sub>▪</sub> ≡ <sub>▪</sub>(∃γ′)(α′ ϵ γ′ <sub>▪</sub> (<i>x</i>′)(<i>x</i>′ ϵ γ′<sub>▪</sub> ⊃ <sub>▪</sub>∼((α′′)(α′′ ϵ <i>x</i>′<sub>▪</sub> ≡ <sub>▪</sub>α′′ ϵ <i>x</i>′))))))))<br />
<br />mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-32774880526226937422015-07-11T17:57:00.001-04:002015-07-11T17:57:55.237-04:00Binary Operators in somerby.net/mack/logicBinary Operators in somerby.net/mack/logic now have different precedences. See <a href="http://somerby.net/mack/logic/documentation.html#language_logical_operators" target="_blank">here</a>. mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-34183594353906275972015-06-21T17:34:00.000-04:002016-05-26T15:29:10.778-04:00Alternative Hexadecimal DigitsI've been collaborating with <span class="wikilink"><a href="http://odo.lv/Blog/150502?language=en" target="_blank">Valdis Vītoliņš</a> on hexadecimal digits. The result is a new set of digits:</span><br />
<span class="wikilink"><br /></span>
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<a href="http://somerby.net/mack/images/HandwrittenHexadecimalDigits2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://somerby.net/mack/images/HandwrittenHexadecimalDigits2.jpg" height="185" width="320" /></a></div>
They follow a design where the horizontal strokes represent 1, 2 and 4 in the binary composition of the number which each digit is supposed to represent. The rules for constructing the digits are:<br />
<ul>
<li>0 is represented by a digit that looks like an 'o' or a '6'.</li>
<li>8 is represented by a digit that looks like a miniscule rho or a 'P'.</li>
<li>Numbers 1-7 and 9-15 are represented by digits whose shape follows this plan: </li>
</ul>
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<a href="http://somerby.net/mack/images/DigitPlan.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://somerby.net/mack/images/DigitPlan.png" /></a></div>
<span class="wikilink">We considered several possible sets of digits before settling on this one. We choose this new set of digits because</span><span class="wikilink"> </span><br />
<ol>
<li><span class="wikilink">We find it is the easiest to encode and decode.</span></li>
<li><span class="wikilink">We find that pairs of these digits can be combined into readable ligatures.</span></li>
</ol>
<span class="wikilink">Valdis has created fonts for the digits and ligatures, which I have incorporated into a branch of the Hex Editor plugin for <a href="https://notepad-plus-plus.org/" target="_blank">Notepad++ </a>. It has all the features of the mainline <a href="http://sourceforge.net/projects/npp-plugins/files/Hex%20Editor/Hex%20Editor%20Plugin%20v0.9.5/" target="_blank">Hex Editor plugin</a>, but also offers the option of viewing hexadecimal data with the new characters in place of the traditional 0-9A-F. If you'd like to use it, then download this zip file and run the setup executable contained therein:</span><br />
<ul>
<li><span class="wikilink"><a href="http://somerby.net/mack/HexEditor.zip">HexEditor.zip</a> </span></li>
</ul>
<span class="wikilink">The fonts look like this:</span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://somerby.net/mack/images/NewLigatureSansTable.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://somerby.net/mack/images/NewLigatureSansTable.png" height="320" width="211" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://somerby.net/mack/images/NewLigatureTable.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://somerby.net/mack/images/NewLigatureTable.png" height="320" width="204" /></a></div>
mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com1tag:blogger.com,1999:blog-35719397.post-48640938339660258852015-05-24T18:52:00.000-04:002015-05-24T18:55:16.727-04:00Hexadecimal Digits AND Ligatures!<br />
A few years ago, I posted a set of hexadecimal digits that I invented that could serve as an alternative to the customary 0-9 and A-F. My idea was to make a set of digits that could be represented in the standard numeric LCD display where the digit somehow represented a binary version of the number it represents. See <a href="http://mackwai.blogspot.com/2009/07/alternative-hexadecimal-digits.html" target="_blank">here</a>. Now someone has done one better. In the blog post "<a href="http://odo.lv/Blog/150502?language=en" target="_blank">We think about yotabytes, but can't handle just one byte</a>", <span class="wikilink">Valdis Vītoliņš describes a hexadecimal of his own invention based on the same idea which adds one innovation; he combines pairs of digits into ligatures, thereby creating a system of 256 symbols that can represent numbers 0 though 255. That's one symbol for each possible byte value, and the numeric value for each symbol can be read out of its shape with a simple algorithm. If you're a programmer <ahem>, <i>software engineer,</i> like me, who looks at <a href="http://en.wikipedia.org/wiki/Hex_dump" target="_blank">hex dumps</a> often, this is an intriguing idea. I might actually try use this.</span>mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-58918851835628118672015-05-03T10:03:00.001-04:002015-05-03T10:03:05.935-04:00Counterexample Feature Completed for somerby.net/logicThe "Counterexample" button now works for modal statements as well as non-modal statements in somerby.net/mack/logic.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-34205917373685242562015-04-14T20:59:00.000-04:002015-04-14T20:59:22.133-04:00Addtions to somerby.net/mack/logicI added operators for <a href="http://somerby.net/mack/logic/documentation.html#language_logical_operators">strict implication</a> and <a href="http://somerby.net/mack/logic/documentation.html#language_definite_description">definite description</a> to <a href="http://somerby.net/mack/logic" target="_blank">somerby.net/mack/logic</a>. I also added the beginnings of a feature that finds counterexamples for statements that are not necessarily true. It only works for non-modal statements, but I think I can expand it to work for modal statements as well.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-83944535859577928462015-03-28T11:47:00.000-04:002015-03-28T11:47:47.040-04:00Free Variables Are Back (partially)I've added some support for free variables back into somerby.net/mack/logic. They are now treated as constants denoting actual objects. At the moment, they are not allowed in statements which contain modal operators, since I have not yet found a satisfactory way to deal with constants in modal statements. Or at least a way that I'm willing to commit to. I'm working on it. I've been reading <a href="http://www.amazon.com/Possible-Worlds-Rod-Girle/dp/0773526684" target="_blank"><i>Possible Worlds</i></a>, "<a href="http://plato.stanford.edu/entries/actualism/index.html" target="_blank">Actualism</a>" and "<a href="http://plato.stanford.edu/entries/actualism/SQML.html" target="_blank">SQML</a>" and other such things for instruction and insight. I knew that there was more than one system of modal logic, but this reading has made me appreciate the need for me to define exactly what system of modal logic somerby.net/mack/logic implements. So I'm working on that, too.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-73625439559227411922015-02-13T19:44:00.000-05:002015-02-13T19:44:26.876-05:00No More Free Variables (for now)Formerly, <a href="http://somerby.net/mack/logic">somerby.net/mack/logic</a> would accept statements with free variables. It would bind them all with existential quantifiers. That seemed to work; it yielded the results I expected for all of the statements I had tried - up until a few days ago, when I found it to fail for some statements. One such statement is "<span style="color: orange;"><span style="font-family: "Courier New",Courier,monospace;">a=b</span></span>". I would expect this to be a contingent statement; true or false depending on whether or not <span style="color: orange;"><span style="font-family: "Courier New",Courier,monospace;">a</span></span> and <span style="color: orange;"><span style="font-family: "Courier New",Courier,monospace;">b</span></span> denoted the same object, but the application decided it to be necessarily true. It's not hard to see why if we expand the statement, as the application does, to be "<span style="color: orange;"><span style="font-family: "Courier New",Courier,monospace;">3b,3a,a=b<span style="font-family: inherit;"></span></span></span>". There is always an <span style="color: orange;"><span style="font-family: "Courier New",Courier,monospace;">a</span></span> and a <span style="color: orange;"><span style="font-family: "Courier New",Courier,monospace;">b</span></span> which are identical in any nonempty universe; they are identical when they are the same object.<br />
<br />
So, for now, the application rejects statements that contain free variables. I'm working on finding a way to decide them correctly. As soon as I've found one (and I've assured myself that it really is correct), I'll incorporate it into the application and allow free variables once again.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-91443483363739558902015-01-24T21:03:00.001-05:002015-02-13T19:08:24.805-05:00Another Square of OppositionIn my former post, I showed Terrence Parsons' theory of Aristotle's Square of Opposition in symbolic form. I also noted that the Square of Opposition holds up under the modern interpretation of the four forms of Term Logic if it is assumed <i>a priori</i> that the subject terms of the forms is non-empty (I am not the first to note this; see <a href="http://www.butte.edu/~wmwu/iLogic/2.2/iLogic_2_2.html" target="_blank">section 2.2.2 of iLogic</a>). So there are two interpretations of the four forms which affirm the Square of Opposition. But are there others?<br />
<br />
I found another interpretation of the four forms of Term Logic that affirms the Square of Opposition. It's a parallel to Parsons' Square. Parson constructs the square by taking the modern interpretation of the four forms, bestowing existential import upon Form A and denying existential import to the form on the opposite corner - Form O. The interpretation I found is constructed by by taking the modern interpretation of the four forms, bestowing
existential import upon Form E, and denying existential import to Form I on the opposite corner. Here is a statement of it in symbolic form:<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// Another Square of Opposition
// "All S are P", with no existential import
A <=> (x,Sx->Px)
// "No S are P", with existential import
E <=> ((x,Sx->~Px)&(3x,Sx))
// "Some S are P", with no existential import
// "If there are any S, some of them are P" might be a better way to state it.
I <=> ((3x,Sx)->(3x,Sx&Px))
// "Some S are not P" under the modern interpretation with existential import
// Since it has existential import, there's no need to state is as "Not all S are P".
O <=> ~(x,Sx->Px)
->
// Contraries
~(A&E)
// Contradictories
A ^ O
I ^ E
// Subcontraries
I | O
// Subalterns
A -> I
E -> O
</pre>
<br />
I'd like know if anyone else has thought of it before.<br />
<br />
This interpretation, like Parsons' interpretation and the modern interpretation combined with a non-empty subject term, affirms <a href="http://en.wikipedia.org/wiki/Logical_hexagon" target="_blank">the Logical Hexagon</a>: <br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// The Logical Hexagon:
// "All S are P", with no existential import
A <=> (x,Sx->Px)
// "No S are P", with existential import
E <=> ((x,Sx->~Px)&(3x,Sx))
// "Some S are P", with no existential import
// "Some S are P, if any S exist" is a better way to state it.
I <=> ((3x,Sx)->(3x,Sx&Px))
// "Some S are not P", existential import
// Since it has existential import, there's no need to state is as "Not all S are P".
O <=> ~(x,Sx->Px)
// The statement U may be interpreted as "Either all S are P or all S are not P."
U <=> ((x,Sx->Px)|(x,Sx->~Px))
// The statement Y may be interpreted as "Some S is P and some S is not P"
Y <=> ((3x,Sx&Px)&(3x,Sx&~Px))
->
// Subalterns: AI, AU, EU, EO, YI, YE
A->I
A->U
E->U
E->O
Y->I
Y->O
// Contraries: AE, EY, YA
~(A&E)
~(E&Y)
~(Y&A)
// Subcontraries: IU, UO, OI
I|U
U|O
O|I
// Contradictories: AO, UY, EI
A^O
U^Y
E^I
</pre>
mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-70310661333463223372015-01-01T19:02:00.000-05:002015-02-13T19:08:44.199-05:00A Second Theory of Term LogicI added Term Logic to <a href="http://somerby.net/mack/logic" target="_blank">somerby.net/mack/logic</a> for fun. While doing the necessary research, I discovered the logical Square of Opposition, which is kind of cool. Terence Parsons wrote an illuminating <a href="http://plato.stanford.edu/entries/square/" target="_blank">article</a> on the Square. In it, he argues convincingly for an interpretation of 2-term propositions that affirms the Square of Opposition, and also convincingly that this interpretation is Aristotle's intended interpretation. I like the article so much that I've chosen to use this interpretation in my application, defining the four forms of propositions (<span style="font-family: "Courier New",Courier,monospace;">SaP</span>, <span style="font-family: "Courier New",Courier,monospace;">SeP</span>, <span style="font-family: "Courier New",Courier,monospace;">SiP</span>, <span style="font-family: "Courier New",Courier,monospace;">SoP</span>) just as he does. Even so, I doubt that this is the only coherent theory of Term Logic held by premodern logicians. Here I shall explain why. You can click on any of the symbolic statements in this post to test them in <a href="http://somerby.net/mack/logic" target="_blank">somerby.net/mack/logic</a>.<br />
<br />
When explaining why the interpretation of the <i>O</i>-form as "Some <i>S</i> is not <i>P</i>" did not cause problems for premodern logicians, Parsons dismisses the possibility that they assumed that the <i>S</i>-term was not empty, stating "Explicitly rejecting empty terms was never a mainstream option, even in the nineteenth century". But I'm not so sure. First of all, just because they did not explicitly reject empty terms does not mean they implicitly rejected empty terms. Second, they did not have to reject empty terms altogether to make this interpretation of <i>O</i>-form compatible with the traditional Square of Opposition. They only needed to assume <i>a priori</i> (and perhaps unconsciously) that the <i>S</i>-term was not empty whenever they were making an argument. This isn't a very rigorous thing to do, but it's a natural thing to do. Usually, if we are making assertions about some kind of thing, it's because some such thing exists and we want to say something meaningful about it. Reasoning about unicorns may have its uses, but they are not obvious.<br />
<br />
Suppose that some pre-modern philosophers, like Boethius and Peter of Spain, did not interpret the propositional forms as Aristotle intended. Instead, they assumed <i>a priori</i> that the <i>S</i>-term was nonempty, and let the <i>O</i>-form have existential import, just as Boethius seemed to be doing when he translated "Some <i>S</i> is <i>P</i>". Then, instead of the Square of Opposition being this:<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// Aristotle's Square of Opposition
A <=> ((x,Sx->Px) & (3x,Sx))
E <=> (x,Sx->~Px)
I <=> 3x,Sx&Px
O <=> ((3x,Sx&~Px)|(~3x,Sx))
->
// Contraries
~(A&E)
// Contradictories
A ^ O
I ^ E
// Subcontraries
I| O
// Subalterns
A -> I
E -> O
</pre>
<br />
they believed the Square of Opposition was this:<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// A Hypothetical Alternative to
// Aristotle's Square of Opposition
3x,Sx // Assume a priori that S is not empty.
A <=> (x,Sx->Px) // (Existential import here would be redundant.)
E <=> (x,Sx->~Px)
I <=> (3x,Sx&Px)
O <=> (3x,Sx&~Px) // Assume O has existential import.
->
// Contraries
~(A&E)
// Contradictories
A ^ O
I ^ E
// Subcontraries
I | O
// Subalterns
A -> I
E -> O
</pre>
<br />
The relationships of the Square hold in this interpretation as well as in Aristotle's.<br />
<br />
And then there is the matter of the Principle of Obversion and the Principle of Contraposition. Parsons says that some medieval logicians advocated these principles, though they are both fallacious under Aristotle's interpretation of the four forms. The following is not necessarily true:<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// The Principle of Conversion by Contraposition,
// with Aristotle's interpretation
// of the A-form and the O-form
((x,Sx->Px) & (3x,Sx)) <=> ((x,~Px->~Sx) & (3x,~Px))
((3x,Sx&~Px)|(~3x,Sx)) <=> ((3x,~Px&~~Sx)|(~3x,~Px))
</pre>
<br />
This is not necessarily true, either:<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// The Principle of Obversion,
// with Aristotle's interpretation
// of the A-form and the O-form
// Every S is P = No S is non-P (SaP <=> Se~P)
((x,Sx->Px) & (3x,Sx)) <=> (x,Sx->~~Px)
// No S is P = Every S is non-P (SeP <=> Sa~P)
(x,Sx->~Px) <=> ((x,Sx->~Px) & (3x,Sx))
// Some S is P = Some S is not non-P (SiP <=> So~P)
(3x,Sx&Px) <=> ((3x,Sx&~~Px)|(~3x,Sx))
//Some S is not P = Some S is non-P (SoP <=> Si~P)
((3x,Sx&~Px)|(~3x,Sx)) <=> (3x,Sx&~Px)
</pre>
<br />
Why did some logicians make these mistakes? And why did other logicians like Peter of Spain endorse them? Maybe to them, they weren't mistakes. Under what we call the modern interpretations of the four forms, these principles are necessarily true.<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// The Principle of Conversion by Contraposition,
// with the modern interpretations of the forms:
(x,Sx->Px) <=> (x,~Px->~Sx)
(3x,Sx&~Px) <=> (3x,~Px&~~Sx)
</pre>
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">// The Principle of Obversion,
// with the modern interpretations of the forms:
// Every S is P = No S is non-P (SaP <=> Se~P)
(x,Sx->Px) <=> (x,Sx->~~Px)
// No S is P = Every S is non-P (SeP <=> Sa~P)
(x,Sx->~Px) <=> (x,Sx->~Px)
// Some S is P = Some S is not non-P (SiP <=> So~P)
(3x,Sx&Px) <=> (3x,Sx&~~Px)
//Some S is not P = Some S is non-P (SoP <=> Si~P)
(3x,Sx&~Px) <=> (3x,Sx&~Px)
</pre>
<br />
Being necessarily true, they will still, of course, be true under an <i>a priori</i> assumption that the <i>S</i>-term is nonempty. So maybe there was a theory of term logic floating around Medieval Europe that looked like this:<br />
<br />
<pre onclick="PassToLogic(this)" style="color: orange; cursor: pointer;">3x,Sx
A <=> (x,Sx->Px)
E <=> (x,Sx->~Px)
I <=> (3x,Sx&Px)
O <=> (3x,Sx&~Px)
->
// Contraries
~(A&E)
// Contradictories
A ^ O
I ^ E
// Subcontraries
I | O
// Subalterns
A -> I
E -> O
</pre>
<br />
If so, then they really did have a coherent theory of Term Logic which affirmed the Principle of Conversion by Contraposition and the Principle of Obversion. I can't be sure, since I haven't looked for evidence to the contrary, e.g. Peter of Spain discussing empty terms in <cite>Summulae Logicales Magistri Petri Hispani</cite>, but as far as I know, it makes sense. I guess I'll have to read some Medieval logic to find out. It's too bad I don't know Latin.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com0tag:blogger.com,1999:blog-35719397.post-24665227523226687962014-11-28T20:55:00.002-05:002015-02-13T19:08:10.178-05:00Feedback for somerby.net/mack/logicAnyone who wishes to comment on <a href="http://somerby.net/mack/logic">somerby.net/mack/logic</a>, please leave your comments here.mackwaihttp://www.blogger.com/profile/08762010168711247828noreply@blogger.com3